Structure Of Ni Rings Related To Centers

We first obtain that NI rings satisfy a property that if ab is central for elements a, b, then (ab)(n) = (ba)(n) for some n >= 1, by applying a property of reduced rings. We prove next the following: Let R be a ring and I be the ideal of R generated by the subset {ab - ba} a, b is an element of R such that ab is central in R}. (i) Suppose that ab is central for a, b is an element of R and ab - ba is a nonzero nilpotent. Then, A(ab - ba)A is a nonzero nilpotent ideal of the subring A of R, where 1 is the identity of R, B = Z . 1 = {n1 vertical bar n is an element of R Zg, and A is the algebra B < a, b > generated by a, b over B. (ii) If R is NI, then I is nil and R/I is an Abelian NI ring. (iii) Let R be reversible and ab be central for a, b is an element of R. Then, there exists l >= 1 such that, for every n >= l, (ab)(n) = (ba)(n) and (ab)(n) = bh (ab)(n-h) a(h) for all 1 <= h <= n; especially a(n)b(n) = (ab)(n) = b(n)a(n). We call a ring pseudo-NI if it satisfies the first property of NI rings to be mentioned and examine the structures of NI and pseudo-NI rings in several ring theoretic situations, showing that semisimple Artinian rings are pseudo-NI.